new amplitude variations of modulated rv tauri stars support the … · 2019. 5. 6. ·...

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Astronomy & Astrophysics manuscript no. main_rev c©ESO 2018November 6, 2018

Amplitude variations of modulated RV Tauri stars support the dust

obscuration model of the RVb phenomenon

L.L. Kiss1, 2 and A. Bódi1, 3

1 Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Hungarian Academy of Sciences, H-1121 Budapest,Konkoly Thege M. út 15-17, Hungarye-mail: [email protected]

2 Sydney Institute for Astronomy, School of Physics A28, University of Sydney, NSW 2006, Australia3 Department of Experimental Physics and Astronomical Observatory, University of Szeged, H-6720 Szeged, Dóm tér 9., Hungary

July 2017

ABSTRACT

Context. RV Tauri-type variables are pulsating post-AGB stars that evolve rapidly through the instability strip after leaving theAsymptotic Giant Branch. Their light variability is dominated by radial pulsations. Members of the RVb subclass show an additionalvariability in form of a long-term modulation of the mean brightness, for which the most popular theories all assume binarity andsome kind of circumstellar dust. Here we address if the amplitude modulations are consistent with the dust obscuration model.Aims. We measure and interpret the overall changes of the mean amplitude of the pulsations along the RVb variability.Methods. We compiled long-term photometric data for RVb-type stars, including visual observations of the American Association ofVariable Star Observers, ground-based CCD photometry from the OGLE and ASAS projects and ultra-precise space photometry ofone star, DF Cygni, from the Kepler space telescope. After converting all the observations to flux units, we measure the cycle-to-cyclevariations of the pulsation amplitude and correlate them to the actual mean fluxes.Results. We find a surprisingly uniform correlation between the pulsation amplitude and the mean flux: they scale linearly with eachother for a wide range of fluxes and amplitudes. It means that the pulsation amplitude actually remains constant when measuredrelative to the system flux level. The apparent amplitude decrease in the faint states has long been noted in the literature but it wasalways claimed to be difficult to explain with the actual models of the RVb phenomenon. Here we show that when fluxes are usedinstead of magnitudes, the amplitude attenuation is naturally explained by periodic obscuration from a large opaque screen, one mostlikely corresponding to a circumbinary dusty disk that surrounds the whole system.

Key words. stars: AGB and post-AGB – stars: oscillations (including pulsations) – stars: variables: general – stars: binaries: general

1. Introduction

The RV Tauri variables constitute a small group of pulsating starswith some dozen known members in the Milky Way and a sim-ilar number of variables in the Magellanic Clouds. They are F,G and K-type supergiants that form the high-luminosity exten-sion of Population II Cepheids in the classical instability strip(Wallerstein 2002). The rarity of RV Tau-type variables can beexplained by their evolutionary state given that these objects arepost-AGB stars rapidly evolving in the Hertzprung-Russell dia-gram blueward from the Asymptotic Giant Branch (AGB) on thetimescales of 103–104 yrs (Blöcker 1995). During this evolution,post-AGB stars cross the instability strip, where they becomepulsationally unstable (Fokin 1994, Fokin et al. 2001, Aikawa2010). The most regular, though far from being Cepheid-like,high-amplitude light curves belong to the RV Tauri stars, whichare located in cooler part of the post-AGB instability strip, be-tween 5000 K and 6000 K (Kiss et al. 2007, Bódi et al. 2016).

The most distinct feature of the RV Tau-type variables is thepresence of alternating minima of the pulsations (meaning thatevery second light curve minimum is shallower), with typical(double) periods from 30 days to 90 days. The periodicity is notstrict, as the cycle-to-cycle variations can be quite significant,and in some cases, it has been shown to originate from low-dimensional chaos (e.g. Buchler et al. 1996). In addition to thepulsations, some RV Tau stars show long-term modulation of the

mean brightness, with periods of 700-2500 days. The absence orpresence of the slow modulation is the basis for classifying thestars into the RVa and RVb photometric subclasses, respectively.

The RVb phenomenon has been connected to the fact thatthese stars are typically redder than the RVa-type variables andthe role of circumstellar dust shells was presumed (Lloyd Evans1985). More recent studies interpret the RVb phenomenon withperiodic obscuration events in binary systems (Fokin 1994, Pol-lard et al. 1996, Van Winckel et al. 1999, Fokin et al. 2001, Maaset al. 2002, Gezer et al. 2015). In that picture all RVb stars arebinaries, surrounded by a large opaque screen, relative to whichthe pulsating component changes its position during the orbit.The presence of a dusty disk is indeed seen in the infrared partof the Spectral Energy Distribution (see Gezer et al. 2015 for thelatest analysis that used WISE data and references therein). Inaddition to the disk, it has also been speculated that interactionsof the components may lead to changes in the pulsation ampli-tude (Pollard et al. 1996, Maas et al. 2002, Pollard et al. 2006),which could further complicate the models of these stars.

The fact that the pulsations in the light curve appear to vi-sually decrease in amplitude in the faint states of some RVbvariables has been regularly noted in the literature. Fokin (1994)compiled data for a dozen stars and noted that all showed sys-tematically lower primary light amplitudes in the minimum ofthe secondary variation. Pollard et al. (1996) discussed in de-

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A&A proofs: manuscript no. main_rev

tails the well-expressed pulsation amplitude decrease in the faintstates of U Mon and AI Sco (i.e. around the RVb-type min-ima), which they assumed to be caused by some sort of dynam-ical interaction between the pulsating and the companion stars,which affects both the pulsations and the mass-loss processes.Pollard et al. (1996) explicitly argued that pure obscuration bydust should not decrease the amplitude of the pulsations. Thisline of argumentation was later adopted by Van Winckel et al.(1999) and Maas et al. (2002), who claimed that the detectedamplitude change is difficult to account for in a simple geomet-ric picture. Later, Pollard et al. (2006) reiterated the main con-clusion that enhanced mass-loss or binary interaction would beneeded around the periastron of U Mon to explain the apparentamplitude damping. Most recently, Percy (2015) investigated theamplitude variability of 42 RV Tau variables and found that thepulsation amplitude can change by factors of up to 10, on mediantime-scales of about 22 pulsation periods. He concluded that thecause of the pulsation amplitude variations remains unknown.

Despite the efforts in the past decades, high-quality photom-etry of RV Tau-type stars is still very rare. In the original Keplerfield, there is only one star, DF Cygni, a moderately bright RVb-type variable, for which independent analyses were recently pub-lished by Bódi et al. (2016) and Vega et al. (2017). Both stud-ies pointed out that DF Cygni shows very rich behaviour on alltimescales. While Bódi et al. (2016) put their emphasis on de-tecting evidence of strong non-linear effects that are directly ob-servable in the Kepler light curve, Vega et al. (2017) used Ke-pler data to argue for binarity as the main cause of long-periodvariability in DF Cyg. The latter authors also noted that whenthe light curve is measured in fluxes, the reduction of the pulsa-tion amplitude in the faint state of DF Cyg is exactly the same∼90% than the overall fading by 90% during the RVb mini-mum. The exact correspondence of the two can be naturally ex-plained by 90% obscuration of the pulsating stellar disk by avery large opaque screen during every orbit, so that the localchanges around the mean (i.e. the short-period pulsations) are infact constant when considered relative to the mean brightness ofthe system.

This idea is the very inspiration of the present paper. We be-lieve the earlier studies overlooked a very simple and yet im-portant point by sticking to the inverse logarithmic magnitudesystem in the variable star analyses, rather than using the physi-cally meaningful flux units, which scale linearly with the photoncounts. As we will demonstrate for the best observed RVb-typevariables, a very convincing and ubiquitous correlation exists be-tween the flux amplitudes and the mean flux levels for all RVbstars, which indicates that dust obscuration, most likely by cir-cumbinary disks, can indeed be the universal explanation for theRVb phenomenon.

2. Data and methods

Given the rarity of the RV Tau-type stars, there are not manyvariables with extensive observations. We have surveyed the TheInternational Variable Star Index (VSX database1) and the lit-erature for well observed and characterized targets. We endedup with three major sources of RVb photomeric data. First, wechecked the database of visual observations of the American As-sociation of Variable Star Observers (The AAVSO InternationalDatabase). In total, we found eight stars with several cycles ofthe RVb variability and duty cycle in excess of 75%, which wasfound to be necessary to measure the amplitudes of individual

1 http://www.aavso.org/vsx/

Table 1. The studied sample of RVb stars. Tobs and Nobs are the time-span and the total number of observations. Ppul and Pmod refer to the pe-riods of pulsation and RVb-type modulation, respectively, as measuredfrom the analysed data. The OGLE variable names were shortened byomitting “OGLE-” in front of the shown identifiers.

Name Tobs Nobs Ppul Pmod Source(d) (d) (d)

IW Car 18120 4685 71.98 1449 AAVSO3300 2179 72.2 1470 ASAS

SX Cen 22409 1320 32.88 602 AAVSO3296 1153 33.01 610 ASAS

DF Cyg 17074 5924 49.82 780 AAVSO1470 66533 49.84 786 Kepler

SU Gem 14783 2228 49.92 682 AAVSOU Mon 46283 48019 91.48 2451 AAVSOAR Pup 14998 1450 76.66 1194 AAVSO

3299 1086 76.34 1178 ASASAI Sco 19538 1408 71.64 977 AAVSORV Tau 40020 14976 78.48 1210 AAVSOBLG-T2CEP-177 2836 742 92.44 2970 OGLEBLG-T2CEP-215 2829 814 55.74 958 OGLEBLG-T2CEP-345 2830 1344 73.64 1100 OGLEBLG-T2CEP-350 2776 1026 87.20 722 OGLEBLG-T2CEP-352 4404 978 103.78 543 OGLEBLG-T2CEP-354 3232 533 66.46 951 OGLELMC-T2CEP-200 4494 917 69.86 850 OGLE

pulsation cycles. Then we checked the online catalogue of theAll Sky Automated Survey (ASAS, Pojmanski 2002) and foundV-band data for three stars. Finally, we surveyed the database ofthe Optical Gravitational Lensing Experiment (OGLE) project,where the OGLE-III Catalog of Variable Stars contains Type IICepheids (including RV Tau-type variables) in the Large Mag-ellanic Cloud (Soszynski et al. 2008), Small Magellanic Cloud(Soszynski et al. 2010) and the Galactic Bulge (Soszynski et al.2011, 2013). Here we found useful I-band data for six stars inthe Bulge and one star in the LMC, all catalogized as ‘RVb’variables by the OGLE team.

The final sample contains 19 datasets for 15 RVb stars, whichis - to our knowledge - the most extensive collection of this kindin the literature. In Table 1 we list the main characteristics ofthe data: the total time-span of the individual light curves, thenumber of measurements per star and two period values: one forthe pulsations and one for the RVb modulation. The listed valueswere measured with standard Fourier analysis, using Period04 ofLenz & Breger (2005). Because of the alternating minima, thehighest peak in the Fourier spectra always corresponded to thehalf-period of the RV Tau cycle, hence we doubled the adoptedperiod of pulsations. The only exception is IW Car, whose lightcurve does not show any alternation in the minima, hence weadopted the single cycle period for its pulsations. The typical un-certainties in the quoted values are in the order of the last digitshown in Table 1. Note that in principle, the pulsation periodcould be determined to more decimals, however, the seeminglyirregular variations in the period/phase (Bódi et al. 2016) wouldpose strong limitations to all future phase predictions. Our mea-sured values are all in good agreement with previous period de-terminations in the literature (e.g. Pollard et al. 1996, Kiss et al.2007, Percy et al. 2015).

The visual data have first been binned to average out the er-rors of the individual observations (typically ±0.2-0.3 mag perpoint). Because of the short pulsation periods, the bin sizes were


L.L. Kiss & A. Bódi: Amplitude variations of RVb stars

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Fig. 1. A gallery of RVb light curves, converted to fluxes and then normalized to unity.



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Fig. 2. A short subset of the AAVSO light curve of U Mon (black dots),overplotted with the fitted linear term (blue lines) and the sum of twosine waves (red dotted lines). The damping of the pulsation amplitudeduring the minimum phase (JD 48,600-48,800) of the RVb variationclearly visible.

selected between 3 days and 5 days, depending the pulsation pe-riod. This way we avoided strong phase smearing due to thebinning, that could have led to undersampled light curves ofpulsations. The CCD observations were only checked for out-liers, identified by visual inspection of the data. Both ASAS andOGLE have daily cadence and no binning was applied to thosedata.

Following the idea outlined by Vega et al. (2017), we con-verted every light curve to flux units with the conversion for-mula f = 10−0.4×(magn.−25), adapting 25.0 as the arbitrary zero-point. To demonstrate the overall continuity and typical dutycycle of the data, in Fig. 1 we show a gallery of the full sam-ple in normalized flux units (unit flux corresponds to the globalmaximum brightness for each star). While we plotted the en-tire ASAS, Kepler and OGLE light curves, the AAVSO dataare only shown partially, in 10,000-days long subsets. For theoverwhelming majority of stars the amplitude variability of theshort-period pulsations is very much apparent, with the highestamplitudes always appearing in the bright phases of the RVbvariability. Exceptions are IW Car, AR Pup (in the ASAS data)and OGLE-BLG-T2CEP-215, which seem to have more stablepulsations with RVb modulations in the same range as the pul-sation amplitudes.

We proceeded in the analysis by measuring the local ampli-tudes of each pulsation cycle. For this, each dataset was splitinto subsets that covered exactly one pulsation cycle (two con-secutive maxima and minima of the alternating depths) and thenwe fitted the following function to each of the subsets:

f (t − t0) = f0 + k(t − t0) +

2∑

n=1

An sin

(

2π(t − t0)

nPpul

+ ϕn

)

(1)

where t and t0 are time and the first time epoch of the subset,respectively, f0, k, An and ϕn (n = 1, 2) are the fitted parameters(for IW Car, n = 1 was used). The zero-point and the linear termwere introduced to include the local variation of the slow RVbmodulation, while the two-component Fourier-polynomial is theapproximation of the light curve shape of the pulsations. The lat-ter is admittedly a simplified description of the light curve, how-

ever, here we are more interested in the global average changesof the pulsation amplitudes than in the perfect fits of the individ-ual cycles. We have also experimented with higher-order poly-nomials, but none of the results changed significantly, hence werestricted the analysis to the simplest form. An illustration of thefitting procedure is shown in Fig. 2, where the red curve showsthe individual fits of Eq. 1 throughout a whole ascending branchof the RVb cycle. The blue line separately indicates the linearterm. The main conclusion here is that the two-component har-monic fit describes the observations very well.

After fitting Eq. 1 to a given subset, we subtracted the linearterm and computed the two extrema of the residual polynomial(except for the Kepler data of DF Cyg, where the extrema weredetermined from the residual of the observational points). Theirdifference was taken as the full pulsation amplitude in the givensubset, while the median of the subset was chosen to representthe average flux of the star. The relationship that was found be-tween these amplitudes and mean fluxes and its properties rep-resent the main results of this paper.

3. Results

We treated the four types of data (visual, OGLE I-band, ASASV-band, Kepler) separately, partly because of the wavelength de-pendence of the pulsation amplitudes (Pollard et al. 1996), partlyto avoid misleading effects from mixing heterogeneous data.

First we show the raw correlations between the mean fluxesand the corresponding full amplitudes in the upper panels ofFigs. 3-6. For the sake of comparing bright and faint stars inFigs. 4-6, we set doubly logarithmic scaling of the axes. Toguide the eye, we have also drawn the lines of equality in theseplots. While Kepler and visual data lie close to the equality, theOGLE and ASAS points all form parallel sequences falling be-low the diagonal lines, meaning that the correlations are closeto linear but the mean slopes are less than one. The formal Pear-son’s r correlation coefficients are greater than 0.8 for most of thestars except those three already mentioned in relation to Fig. 1(IW Car: r = 0.53 (AAVSO), AR Pup: r = 0.21 (AAVSO),OGLE-BLG-T2CEP-215: r = 0.34).

The well expressed linear correlation between the ampli-tudes aλ and mean fluxes fλ for all types of the observationsled us to fit the data in the upper panels of Figs. 3-6 withthe simplest aλ = α fλ + β (λ=visual, V and I) linear term.What we found is that the α slope parameters for almost allstars agreed within the error bars for each λ. Namely, the vi-sual observations show a slope parameter around unity (e.g.AI Sco: 0.95±0.10; RV Tau: 0.89±0.08; SU Gem: 1.21±0.14;U Mon: 0.93±0.06), the OGLE I-band observations around 0.6(e.g. BLG-177: 0.59±0.06; BLG-215: 0.68±0.09; LMC-200:0.60±0.04), while the ASAS V-band data resulted in a mean-ingful correlation for SX Cen only, with a slope of 0.62±0.08.The Kepler data of DF Cygni show very similar linear amplitudevs. flux scaling throughout the whole RVb cycle than what thevisual data imply. As we will demonstrate later, the wavelengthdependence of the slope parameter follows that of the intrinsicpulsation amplitude (i.e. the smaller I-band slope parameter is adirect consequence of the pulsation amplitude being smaller in Ithan in V).

In the next step we determined the relative flux changeand the relative pulsation amplitude change as follows. For thefluxes, we defined their relative change as ( f max

λ− fλ)/ f max

λ,

where f maxλ

is the pulsation-averaged flux when the star is thebrightest. This has defined the full range of relative flux changes,where 0.0 corresponds to the global maximum. To calculate the



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Fig. 3. Pulsation amplitudes as function of the mean brightness for theKepler data of DF Cyg. Top panel: instantaneous flux amplitudes vs.mean fluxes in linear scaling. The dotted red line shows a linear fit to thedata, which was used to estimate the f max

λand amax

λparameters. The for-

mal error bars are smaller than the symbol sizes. Bottom panel: the rel-ative pulsation amplitude vs. the relative mean flux in equidistant bins.Here the vertical error bars show the standard deviations of the mean,while the horizontal error bars indicate the width of the bins (note themissing bin at 0.7 due to lack of points). The diagonal black lines showthe line of equality in both panels. See text for the details.

relative amplitude changes, we had to take into account the in-trinsic cycle-to-cycle variability of the RV Tau light curve shapes(well documented for RVa-type stars, e.g. R Sct – Buchler et al.1996, AC Her – Kolláth et al. 1998). Because of that, it wouldhave been misleading to take the single amplitude value that cor-responds to the brightest flux point as the maximum amplitude;instead of that we fitted a line to the amplitude vs. flux plot andthen took the fit’s value for the largest flux as the hypotheticmean maximum amplitude amax

λ(see the upper panel of Fig. 3

for a visualisation of these two parameters). Then the relativechange of the amplitude was defined similarly to that of the flux,

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Fig. 4. Pulsation amplitude as function of the mean brightness for theeight RVb stars from the AAVSO database (for each star, one point cor-responds to one full pulsation cycle). Note the logarithmic scaling inthe upper panel which allows easy comparison of the bright and thefaint stars. The bottom panel is the same as in Fig. 3.

namely as (amaxλ− aλ)/a

maxλ

. Finally, we calculated the mean rel-ative amplitude change as a function of ten equally spaced fluxbins that were selected between the minimum and maximum rel-ative flux changes.

The results are shown in the lower panels of Figs. 3, 4, 5 and6 for the Kepler, AAVSO, OGLE and ASAS data, respectively.Here (0,0) corresponds to the brightest state with the largest am-plitude. Note that the reason why generally there is no point inthe origin is twofold: (i) the bins are centered on their midpointsand (ii) the definition of the maximum amplitude implies indi-vidual points in the bright states to scatter around the origin.Two major conclusions can be drawn from these plots. First,within the error bars, the relative amplitude changes scale per-fectly with the relative flux changes, which means that the pul-sation amplitude, in fact, remains constant throughout the RVbcycle when compared to the overall system flux. The diagonal



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Fig. 5. The same as in Fig. 4 for the seven OGLE stars.

lines in the panels are not fits but indicate the lines of equality.All but a few RVb stars show the same proportionality than theone found for DF Cygni by Vega et al. (2017) and this scalingis valid not only for the extrema (what actually was noticed bythose authors) but it holds throughout the whole RVb cycle. Sec-ond, the scatter in the plots is rather dominated by the stars thanby the observational uncertainties. This is illustrated by the Ke-pler plots in Fig. 3, where the individual points in the upper panelhave practically no measurement errors. It is the RV Tau-typepulsation that changes seemingly irregularly (presumably due tostrong non-linear effects, Bódi et al. 2016) in the same range thanthe apparent scatter around the diagonal lines in Figs. 3-6. Whatthese mean is that the simplest explanation shall invoke a mech-anism that equally affects the mean brightness and the apparentamplitude.

4. Discussion

Our results show a general pattern for the overwhelming major-ity of the sample, that is a linear correlation between the mean

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Fig. 6. The same as in Fig. 4 for the three ASAS stars.

brightness and the pulsation amplitude, when everything mea-sured in flux units. The one-to-one correspondence of the rel-ative changes, depicted in the lower panels of Figs. 3-6, is ac-tually rephrasing the linear correlation differently, which can beunderstood as follows. Let us consider the equality of the relativechanges of the pulsation amplitudes and the mean fluxes

∆aλ

aλ=∆ fλ

fλ(2)

as an approximation of the following differential equation

daλ

aλ=

d fλ

fλ. (3)

This can be easily integrated as

log aλ = log fλ + cλ, (4)



where the integration constant cλ can be expressed as

cλ = logamaxλ

f maxλ

. (5)

Here amaxλ

and f maxλ

are the amplitude and the mean flux valueswhen the star is the brightest. Substituting cλ into Eq. 4 and re-arranging leads to the following formal solution:

aλ =amaxλ

f maxλ

fλ, (6)

which is exactly the same linear relationship between the ampli-tudes and the fluxes as we have seen in the empirical data. Theα slope parameter turns out to be the ratio of the maximal am-plitudes and fluxes, which can be determined for each star andeach photometric band separately. Given that the RV Tau pulsa-tions follow the same wavelength dependence in the photometricamplitudes as, for example, the Cepheid variables, the smallerI-band slope parameters found for the OGLE-stars are a directconsequence of having smaller pulsation amplitudes in I than inV (e.g. Pollard et al. 1996).

Having established these very simple properties of the flux-amplitude relationship, one can ask about the implications. Ithas been noticed very early for individual stars that the pul-sation amplitude shows some correlation with the long-termchanges (for instance, O’Connell 1946 already noted for IW Carthat the pulsations have larger amplitudes when the star wasbrighter). A connection between the circumstellar material andthe RVb phenomenon was proposed by Lloyd Evans (1985)who argued against the role of binarity, preferring unstable, po-tentially pulsation-induced mass-loss that can lead to R Coro-nae Borealis-like obscuration events. Fokin (1994) consideredthe problem of the secondary variability in RVb stars and, afterdiscussing the difficulties of modelling it with some sort of pul-sations, concluded that binarity shall play an important role in-stead, with quasi-periodic eclipses by a circumstellar dust cloud(similarly to the proposal of Waelkens & Waters 1993). Zsol-dos (1996) too argued against pulsational origin of the RVb phe-nomenon in RV Tau but noted that simple binarity is also ex-cluded because of the long-term changes of the RVb cycles. Pol-lard et al. (1996), based on extensive long-term multicolour pho-tometry of RV Tau stars, noted various features of the RVb phe-nomenon. Two of their stars, U Mon and AI Sco, exhibited well-expressed ‘amplitude damping’ during the RVb minima. Whilebinarity was suggested for these stars, they also noted as an ar-gument against dust obscuration that “pure obscuration by dustwill give a reddening and a dimming of the obscured star butshould not decrease the amplitude of the pulsations or make thedeep-shallow alternations less distinct”. This line of argumenta-tion was adopted by later authors, like Van Winckel et al. (1999)and Maas et al. (2002).

We think it is clear that the recurrent reasoning in the litera-ture against dust obscuration is not correct. It was only very re-cently, that Vega et al. (2017) pointed out that this kind of ampli-tude damping is actually what one expects for the pulsating stel-lar disk being occulted during the long-period minima by a verylarge, opaque screen, which, they argue, should be a circumbi-nary dusty disk around the entire binary system. Our results pre-sented in this paper indicate that amplitude variability is ubiqui-tuous and follows the same linear scaling with the mean flux inalmost every RVb star. In other words, the pulsation amplituderemains constant (with some intrinsic small-scale variability due

to the non-linear nature of the pulsations), when measured rel-ative to the overall system brightness. One of the reasons whythis has not yet been found is the general use of the magnitudesystem in variable star analyses: the inverse logarithmic naturehid the simple relationship between the mean brightness and thepulsation amplitude. Also, the fluctuating amplitude variabilityof RV Tau-type stars did not help either: one needs very goodduty cycle and long time-span to average out the effects of thecycle-to-cycle changes that are inherent to the RV Tau-type pul-sations.

Finally, we briefly turn to those stars that have marginallysimilar behaviour than the whole sample. These are IW Car,AR Pup and OGLE-BLG-TCEP2-215. The classification ofRV Tau stars have long been known to be very difficult (e.g. Zsol-dos 1998), with semiregular (SR) variables frequently mistakenfor other types of luminous variables. The periods, amplitudesand systematic amplitude changes of SR variables (Kiss et al.2000) can indeed by similar photometrically, hence further infor-mation is always important. For IW Car, a rotating and expand-ing post-AGB nebula was recently resolved by ALMA (Bujarra-bal et al. 2017), which is the latest update to its long known post-AGB status. However, its missing light curve alternation meansthat the star is more like a general pulsating post-AGB star thana classical RV Tau-like variable. This is in line with the fact thatGiridhar et al. (1994) derived a spectroscopic effective tempera-ture of 6700 K, which is much hotter than typical RV Tau stars(Kiss et al. 2007). AR Pup and its post-AGB disk was observedinterferometrically by Hillen et al. (2017), who noted that for thisstar, the total infrared luminosity dominates over the dereddenedoptical fluxes, which is indicative of the disk close to edge-on.The least is known for the OGLE star, for which only the veryred colour (V − I ≈ 3.3 mag) is listed in the catalogues, whichcould indicate both high interstellar reddening and instrinsicallyred colour. We speculate that these three stars may have differ-ent geometry and/or circumstellar extinction than the rest of thesample. Even for IW Car and AR Pup, there are cycles of theRVb variability when there are hints of the amplitudes followingthe variations of the mean flux levels. The RVb cycles are notstrictly repetitive in other stars either (Zsoldos 1996), implyingthat the obscuring clouds are changing over the time-scale of theorbital periods of the systems. All in all, while these stars exhibita somewhat noisier relationship than the other stars, fundamen-tally they still exhibit a similarly linear relation between pulsa-tion amplitude and system flux, and therefore fit the proposedgeneral picture.

5. Summary

The main results of the paper can be summarized as follows:

1. We have compiled light curves for a sample of RVb-typevariables, using visual observations, ground-based CCDphotometric measurements and ultra-precise data from theKepler space telescope.

2. We found a ubiquitous linear correlation between the pulsa-tion amplitude and the mean brightess, when both are mea-sured in flux units. There is a one to one correspondance be-tween their relative changes, meaning that the pulsation am-plitude actually remains constant throughout the RVb cycle,when measured relative to the system flux level.

3. The properties of the correlation can be naturally explainedby a mechanism that equally affects the mean flux and the ap-parent amplitude, so that the whole light curve is scaled by atime-dependent factor. Periodically variable obscuration by



a large opaque screen, presumably corresponding to a cir-cumbinary dust disk, provides the required mechanism.

4. We conclude that the light variations of RVb-type stars canbe fully explained phenomenologically by the combinationof time-dependent non-linear pulsations and the dust obscu-ration model of the RVb phenomenon.

Acknowledgements. This work has been supported by the NKFIH K-115709 andthe GINOP-2.3.2-15-2016-00003 grants of the Hungarian National Research,Development and Innovation Office, and the Hungarian Academy of Sciences.This research has made use of the International Variable Star Index (VSX)database, operated at AAVSO, Cambridge, Massachusetts, USA. We acknowl-edge with thanks the variable star observations from the AAVSO InternationalDatabase contributed by observers worldwide and used in this research. This pa-per includes data collected by the Kepler mission. Funding for the Kepler missionis provided by the NASA Science Mission directorate. We thank an anonymousreferee for his/her useful comments and suggestions.

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